There are a couple different ways to think about this. One is purely mathematical, the electric field is defined as being the change in potential:

[tex]\vec{E} = - \nabla V[/tex]

This isn't terribly intuitive though. A more physical way of thinking about it is realising that equipotential surfaces are surfaces where the charge will have the same potential energy. The charge can move about on a given equipotential surface without changing its potential energy and therefore without needing to change its kinetic energy in order to conserve energy. When the charge leaves the equipotential surface its potential energy changes and therefore its kinetic energy must change, thus requiring a force.

Why is the direction of gravity's Force radial (inward, downward)
while gravitational PE depends only on the coorsdinate "r"?
Places at the same potential are connected by paths of zero integrated Field, as in "zero Work", as in F perp. dx